Using a vegetable peeler, shave two long, thin ribbons from cucumber; set aside. Cut a 10cm piece from remaining cucumber and coarsely chop. Muddle with mint leaves, sugar, and lime in a cocktail shaker. Add gin and lime juice and fill shaker with ice. Cover and shake until outside of shaker is frosty, about 20 seconds.

Strain into an ice-filled Collins glasses. Top off each with club soda. Gently stir to combine; garnish with two reserved cucumber ribbons and remaining lime.